Harmonic distortion represents one of the strongest limiting factors on the behaviour of high-quality, analogue microelectronic circuits and other electronic circuits, such as analogue/digital converters (ADC), digital/analogue converters (DAC), filters, etc. Harmonic distortion generally occurs as a result of interference whose level is dependent on the respective signals. In particular, harmonic distortion occurs as a result of the non-linear behaviour of active electronic components (transistors, amplifiers, etc.) or passive electronic components (resistors, capacitors, etc.).
FIG. 1 schematically illustrates a non-linear element 4 with the analogue or digital input signal x(t) or x(n), respectively, and with the analogue or digital output signal y(t) or y(n), respectively. The image of the analogue input signal x(t) on the analogue output signal y(t) is described by the function f, which can be written as a polynomial in the following form:y(t)=f(x(t))=a0+a1x(t)+a2x2(t)+a3x3(t)+,  (1)where a0, a1, a2, a3 . . . are constant coefficients. For simplificity, the following text considers third-order polynomials (ai=0 for i=0, 4, 5, . . . ) with the above expression (a1=1) being normalized, which does not restrict the general applicability of the analysis. If the input signal is a sinusoidal oscillation in the form x(t)=cos(ωt), then Equation (1) results in:
                                          y            ⁡                          (              t              )                                =                                                    1                2                            ⁢                              a                2                2                                      +                                          [                                  1                  +                                                            3                      4                                        ⁢                                          a                      3                      3                                                                      ]                            ⁢                              cos                ⁡                                  (                                      ω                    ⁢                                                                                  ⁢                    t                                    )                                                      +                                          1                2                            ⁢                              a                2                2                            ⁢                              cos                ⁡                                  (                                      2                    ⁢                    ω                    ⁢                                                                                  ⁢                    t                                    )                                                      +                                          1                4                            ⁢                              a                3                3                            ⁢                              cos                ⁡                                  (                                      3                    ⁢                    ω                    ⁢                                                                                  ⁢                    t                                    )                                                                    ,                            (        2        )            where ω is the circular frequency of the sinusoidal input signal. In addition to the linear component, the output signal contains a constant offset as well as second and third harmonics. If the polynomial in Equation (1) also takes account of higher-order terms, then the output signal additionally includes the corresponding higher harmonics.
The aim is to minimize the non-linearity of electronic circuits, that is to say the magnitudes of the coefficients of non-linear order a2, a3, . . .
One possible way to reduce non-linearities is to use high-linearity components for the design of the electronic circuits, for example high-linearity passive components or high-gain amplifiers together with a feedback mechanism.
Another possible way to minimize the non-linear components in electronic circuits is so-called differential technique. This essentially comprises the design of two parallel, ideally identical, signal paths, one of which processes a signal x+(t), while the other processes the inverted version of the signal x−(t)=−x+(t). The input signal x(t) is formed by the difference between the signals in the two signal paths. Thus, x(t)=x+(t)−x−(t)=2x+(t), that is to say the input signal x(t) corresponds to twice the signal x+(t) in the positive signal path. Based on the assumption of the non-linear relationship between the input signal and output signal as described in Equation (1) and the simplification a0=0, this results in the output signals in the positive and negative signal paths y+(t) and y−(t) respectively being:
                                                                           y                +                            ⁡                              (                t                )                                      =                                                            a                  1                                ⁢                                                      x                    +                                    ⁡                                      (                    t                    )                                                              +                                                a                  2                                ⁢                                                      x                    +                    2                                    ⁡                                      (                    t                    )                                                              +                                                a                  3                                ⁢                                                      x                    +                    3                                    ⁡                                      (                    t                    )                                                              +              …                                ⁢                                          ⁢          and                                      (          3          )                                                                                                                              y                    -                                    ⁡                                      (                    t                    )                                                  =                                ⁢                                                                            +                                              a                        1                                                              ⁢                                                                  x                        -                                            ⁡                                              (                        t                        )                                                                              +                                                            a                      2                                        ⁢                                                                  x                        -                        2                                            ⁡                                              (                        t                        )                                                                              +                                                            a                      3                                        ⁢                                                                  x                        -                        3                                            ⁡                                              (                        t                        )                                                                              +                  …                                                                                                        =                                ⁢                                                                            -                                              a                        1                                                              ⁢                                                                  x                        +                                            ⁡                                              (                        t                        )                                                                              +                                                            a                      2                                        ⁢                                                                  x                        +                        2                                            ⁡                                              (                        t                        )                                                                              -                                                            a                      3                                        ⁢                                                                  x                        +                        3                                            ⁡                                              (                        t                        )                                                                              +                  …                                                                                          (          4          )                                                          The            ⁢                                                  ⁢            differential            ⁢                                                  ⁢            output            ⁢                                                  ⁢            signal            ⁢                                                  ⁢                          y              ⁡                              (                t                )                                      ⁢                                                  ⁢            is            ⁢                                                  ⁢            described            ⁢                                                  ⁢            by            ⁢                          :                                ⁢                                          ⁢                                    y              ⁡                              (                t                )                                      =                                                                                y                    +                                    ⁡                                      (                    t                    )                                                  -                                                      y                    -                                    ⁡                                      (                    t                    )                                                              =                                                2                  ⁢                                      a                    1                                    ⁢                                                            x                      +                                        ⁡                                          (                      t                      )                                                                      +                0                +                                  2                  ⁢                                      a                    3                                    ⁢                                                            x                      +                      3                                        ⁡                                          (                      t                      )                                                                      +                0                +                …                                                                          (          5          )                    In this case, the even-numbered order terms (a2, a4, . . . ) cancel one another out. The even-numbered order harmonics of sinusoidal input signals therefore ideally disappear.
This frequently used technique has a number of disadvantages, however:    (1) Odd-numbered harmonics do not cancel one another out.    (2) The cancellation is dependent on matching between the two signal paths. A mismatch of the signals y−(t) and y+(t) in the two signal paths leads to incomplete cancellation of equivalent polynomial coefficients of the even-numbered order terms, and the output signal still includes the corresponding even-numbered harmonics.    (3) Differential circuits lead to higher circuit design costs and occupy more space and chip area, since the signal path must be duplicated, in particular all the passive elements which are used as feedback elements for amplifiers. Furthermore, common-note feedback circuits must be added.